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Have a proof re. cyclic groups, need a little explaining

  1. Mar 3, 2010 #1
    1. The problem statement, all variables and given/known data
    Let a,b be elements of a group G. show that if ab has finite order, then ba has finite order.

    2. Relevant equations

    3. The attempt at a solution
    provided proof:
    Let n be the order of ab so that (ab)n = e. Multiplying this equation on the left by b and on the right by a, we find that (ba)n+1 = bea = (ba)e. Cancellation of the first factor ba from both sides shows that (ba)n = e, so the order of ba is < n. If the order of ba were less than n, a symmetric argument would show that the order of ab is less than n, contrary to our choice of n. Thus ba has order n also.

    okay. so, this is from a chapter on cyclic groups, so i'm assuming that it has to do with, duh, cyclic groups. i know cyclic groups are abelian, and if that were to be assumed from the beginning, i believe this problem would be relatively easy. although, actually, it probably wouldn't even be a problem, since it would be true from the beginning (since we could just say ab=ba, and abn=ban

    Let n be the order of ab so that (ab)n = e.
    got that, that makes sense, i believe that's the definition of order of an element.
    WAIT. here's the crux, something i (maybe) just realized as i typed this sentence. since we define the order of ab such that abn=e, does that imply that G is cyclic, and thus abelian? that would make things easier, if i'm right. so far, a quick internet search has failed me on an answer.
  2. jcsd
  3. Mar 3, 2010 #2


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    If the group is abelian then ab=ba and there is really nothing to prove. The proof you gave holds regardless of whether the group is abelian or not.
  4. Mar 3, 2010 #3
    okay, i was just confused as how b(ab)na could turn into (ba)n+1 without it being abelian. i see how it's from letting (ab)n = e. thanks.
  5. Mar 3, 2010 #4


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    It's actually just from regrouping the terms.
  6. Mar 3, 2010 #5
    yeah, actually, i still don't get it. how can you regroup the terms if it's not commutative? or at least it's not assumed to be commutative.
    we have (ab)(ab)...(ab) n times. so b(ab)na = b(ab)(ab)...(ab)a...
    Oh! got it! thanks!
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